Multi-tone modulation is the basis of the DMT version of ADSL as well as multi-carrier versions of VDSL. This type of modulation is sometimes called orthogonal frequency division multiplexing (ODFM). DMT has more than one encoder and each encoder receives a set of bits that are encoded using a constellation encoder. The output values are the cosine and sine waves and a different cosine and sine frequency is used for each constellation encoder. All the sine and cosine waves are summed together and sent over the channel. Equalization is done by adaptive filters to optimize or nearly optimize transmission. Anytime a channel's frequency response is not flat over the range of frequencies being transmitted inter symbol interference (ISI) occurs.
A per tone equalization for DMT-based systems has recently been proposed. See article by Katleen Van Acker, Geert Leus, Marc Moonen, Oliver van de Weil, and Thierry Pollet, “Per Tone Equalization for DMT Based Systems,” IEEE Transactions Communications, vol. 49, No. 1 pp 109-119, January 2001. The method designs an optimum equalizer for each individual tone and has the potential of achieving an optimum design for DMT based modem systems. The method is summarized by the following.
The traditional DMT modem receiver is first formulated mathematically. To simplify the discussion, it is assumed that the time domain equalizer (TEQ) operates on DMT frames that are perfectly aligned and the length of TEQ and the size of the cyclic prefix are the same. The time domain equalizer removes inter symbol interference from the channel that is longer than the cyclic prefix. The cyclic prefix adds the last L sample points of the 512-point time-domain vector to the beginning of the vector. The interference would otherwise cause a symbol to interfere with the next symbol in time. The time domain equalizer also is used to band pass filter the incoming signal and filter out the out-of-band energy. Let y represent a perfectly aligned DMT frame with cyclic prefix as follows:y=[y0, . . . , y1, . . . , yN+v−2, yN+v−1],  (1)                where N is the size of regular DMT framed and v is the size of cyclic prefix. The standard receiver with time equalization (TEQ) can be represented as follows,        
                                          [                                                                                z                    1                                                                                                ⋮                                                                                                  z                    N                                                                        ]                    =                                                    (                                                                                                    D                        1                                                                                    0                                                              …                                                                                                  0                                                              ⋰                                                              0                                                                                                  ⋮                                                              0                                                                                      D                        N                                                                                            )                            ·                              F                N                                      *                          (                              Y                *                                  w                  teq                                            )                                      ⁢                                  ⁢        where                            (        2        )                                Y        =                              [                                                                                y                                          t                      ,                      o                                                                                                                                  y                                              t                        ,                        1                                                              ⁢                                                                                  ⁢                    …                                                                                        y                                          t                      ,                                              v                        -                        1                                                                                                                  ]                    =                      (                                                                                y                    v                                                                                        y                                          v                      -                      1                                                                                        …                                                                      y                    1                                                                                                                    v                                          v                      +                      1                                                                                                            y                    v                                                                    …                                                                      y                    2                                                                                                                                                                                        ⋰⋰                                                                                                                                                                                                                                                                                  y                                          N                      +                      v                      -                      1                                                                                                            y                                          N                      +                      v                      -                      2                                                                                        …                                                                      y                    N                                                                        )                                              (        3        )            and wteq=[w0 w1 . . . wv−1]T is real v-tap TEQ. FN is N×N Fast Fourier Transform (FFT) matrix., Di the complex 1-tap frequency equalization (FEQ) for tone i, and Y is an N×v Toepliz matrix which contains exactly the same received signal samples as vector y in equation (1). For a single tone, one can rewrite the equation (2) as:Zi=Di*rowi(FN)*(Y*wteq)=rowi(FN*Y)*wteq*Di=rowi(FN*Y)wfeq,i  (4)
By putting Di to the right, one has wteqDi=(wfreq,i)vxl which is a complex v-tap for tone i. The next step then is to allow each tone to have its own optimal v-tap complex frequency equalization (FEQ) as proposed in the cited Van Acker et al. reference.
A per tone equalizer needs to compute a series of Fast Fourier Transforms (FFTs) for each DMT frame as shown in equation (4). Because for each subsequent FFT its time domain samples is only one sample different from its previous FFT, only the very first FFT needs to be computed. The subsequent FFTs can be efficiently calculated by sliding FFT as discussed in the cited Van Acker et al. reference.
Because each tone has its own equalizer, each filter can be optimally designed using minimum mean squared estimation (MMSE) method which is known in the art. The DMT receiver based on per tone equalizer has the potential to achieve optimum performance. Simulation results show that it outperforms almost all the time equalization (TEQ) design algorithms.
Although a per tone equalizer as proposed by the cited Van Acker et al. reference can potentially achieve optimum equalization performance, it is very complex to implement. Each equalizer is a multi-tap complex filter. It demands both high computational complexity and consumes a large amount of data memory.